By Neil D. Jones
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Prove that there is a unary function/: A* -> A* which has no effectively computable time bound. 2. TURING MACHINES—PRELIMINARY DEFINITIONS In order to give the results of the previous section a more rigorous foundation, it is necessary to adopt a precise definition of the terms "effective process," "effectively decidable," and so forth. " In particular the results of the previous section (called "propositions") will all be seen to have close analogs in terms of "Turing decidable," "Turing computable," and so forth.
PROPOSITION II. } is countable. PROOF Each such function must possess an algorithm for computing it. An algorithm is merely a string of symbols over an alphabet consisting of English letters and mathematical and punctuation symbols. We know 1. Implications of the Concept of Effectiveness 33 that the set of all strings over a given alphabet is countable, so the set of all algorithms is countable; hence the set of all effectively computable functions must be countable as well. 2 Effectively Enumerable and Effectively Decidable Sets In what way can we apply the idea of "effective process" to the problem of definition of sets ?
Second, the process must be one which can be performed in a purely finitary manner, one step at a time. It must be deterministic, so that at each step there is at most a single next step to perform (that is, no "guessing" is required). It cannot require the completion of an infinite number of steps, or a continuous series of steps. In addition, each step must itself be purely finitary, so a single step cannot depend upon an infinite amount of data for its completion, or the completion of an infinite process.